We use standard notation for \(L^p\) , \(W^{p,q}\) , and \(C_\textup {b}\) . Moreover, \(C_\textup {c}^\infty \)  is the space of smooth functions with compact support, and \(C([0,T];L_\textup {loc}^p(\mathbb {R}^N))\)  the space of measurable functions \(\psi :[0,T]\rightarrow L_\textup {loc}^p(\mathbb {R}^N)\)  such that \(\psi (t)\in L_\textup {loc}^p(\mathbb {R}^N)\)  for every \(t\in [0,T]\) , \(\sup _{t\in [0,T]}\Vert \psi (t)\Vert _{L^p(K)}<\infty \) , and \(\Vert \psi (t)-\psi (s)\Vert _{L^p(K)}\rightarrow 0\)  when \(t\rightarrow s\)  for all compact \(K\subset \mathbb {R}^N\)  and \(t,s\in [0,T]\) . In a similar way we also define \(C([0,T];L^p(\mathbb {R}^N))\) . Note that the notion of \(\mathbb {R}^N\times (0,T)\ni (x,t)\mapsto \psi (x,t)\in C([0,T];L_\textup {loc}^p(\mathbb {R}^N))\)  is a subtle one. In fact, we mean that \(\psi \)  has an a.e.-version which is continuous \([0,T]\rightarrow L_\textup {loc}^p(\mathbb {R}^N)\) . See e.g. p. 726 in [1]} for more details. The space \(C^{0,\vartheta }(\mathbb {R})\)  with \(\vartheta \in (0,1]\)  is identified as the Banach space with norm \(\Vert \cdot \Vert _{C^{0,\vartheta }(\mathbb {R})}:=\Vert \cdot \Vert _{C_\textup {b}(\mathbb {R})}+|\cdot |_{C^{0,\vartheta }(\mathbb {R})}\)  where
\(|\psi |_{C^{0,\vartheta }(\mathbb {R})}:=\sup _{x,y\in \mathbb {R}}\left\lbrace \frac{|\psi (x)-\psi (y)|}{|x-y|^{\vartheta }}\right\rbrace .\) 
